(a) If A = {p, q, r, s), B = (s, t, u) and C = (s, t, u, v, w}

Find (i) A ∪ (B ∪ C) (ii) (A ∪ B) ∪ C (iii) A ∩ (B ∩ C) (iv) (A ∩ B) ∩ C

(b) If A = {1, 4, 9, 16}, B = {3, 4, 5} and C = {3, 9, 12}

Verify (i) A ∪ (B ∪ C) = (A ∪ B) ∪ C (ii) A ∩ (B ∩ C) = (A ∩ B) ∩ C

(c) If A = {5, 7, 9}, B = {7, 9, 11} and C = {9, 11}

Verify (i) A ∪ (B ∪ C) = (A ∪ B) ∪ (A ∪ C), (ii) A ∩ (B ∩ C) = (A ∩ B) ∩ (A ∩ C)

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Solution

(a) A = {p, q, r, s}, B = {s, t, u} and C = {s, t, u, v, w}

(i) A ∪ (B ∪ C)

B ∪ C = {s, t, u} ∪ {s, t, u, v, w} = {s, t, u, v, w}

∴ A ∪ (B ∪ C) = {p, q, r, s} ∪ {s, t, u, v, w} = {p, q, r, s, t, u, v, w}

(ii) (A ∪ B) ∪ C

A ∪ B = {p, q, r, s} ∪ {s, t, u} = {p, q, r, s, t, u}

∴ (A ∪ B) ∪ C = {p, q, r, s, t, u} ∪ {s, t, u, v, w} = {p, q, r, s, t, u, v, w}

(iii) A ∩ (B ∩ C)

B ∩ C = {s, t, u} ∩ {s, t, u, v, w} = {s, t, u}

∴ A ∩ (B ∩ C) = {p, q, r, s} ∩ {s, t, u} = {s}

(iv) (A ∩ B) ∩ C

A ∩ B = {p, q, r, s} ∩ {s, t, u} = {s}

∴ (A ∩ B) ∩ C = {s} ∩ {s, t, u, v, w} = {s}

(b) A = {1, 4, 9, 16}, B = {3, 4, 5} and C = {3, 9, 12}

(i) B ∪ C = {3, 4, 5} ∪ {3, 9, 12} = {3, 4, 5, 9, 12}

∴ A ∪ (B ∪ C) = {1, 4, 9, 16} ∪ {3, 4, 5, 9, 12} = {1, 3, 4, 5, 9, 12, 16}

A ∪ B = {1, 4, 9, 16} ∪ {3, 4, 5} = {1, 3, 4, 5, 9, 16}

∴ (A ∪ B) ∪ C = {1, 3, 4, 5, 9, 16} ∪ {3, 9, 12} = {1, 3, 4, 5, 9, 12, 16}

Hence, A ∪ (B ∪ C) = (A ∪ B) ∪ C

(ii) B ∩ C = {3, 4, 5} ∩ {3, 9, 12} = {3}

∴ A ∩ (B ∩ C) = {1, 4, 9, 16} ∩ {3} = Φ

A ∩ B = {1, 4, 9, 16} ∩ {3, 4, 5} = {4}

∴ (A ∩ B) ∩ C = {4} ∩ {3, 9, 12} = Φ

Hence, A ∩ (B ∩ C) = (A ∩ B) ∩ C

(c) A = {5, 7, 9}, B = {7, 9, 11} and C = {9, 11}

(i) B ∪ C = {7, 9, 11} ∪ {9, 11} = {7, 9, 11}

∴ A ∪ (B ∪ C) = {5, 7, 9} ∪ {7, 9, 11} = {5, 7, 9, 11}

A ∪ B = {5, 7, 9} ∪ {7, 9, 11} = {5, 7, 9, 11}

A ∪ C = {5, 7, 9} ∪ {9, 11} = {5, 7, 9, 11}

∴ (A ∪ B) ∪ (A ∪ C) = {5, 7, 9, 11} ∪ {5, 7, 9, 11} = {5, 7, 9, 11}

Hence, A ∪ (B ∪ C) = (A ∪ B) ∪ (A ∪ C)

(ii) B ∩ C = {7, 9, 11} ∩ {9, 11} = {9, 11}

∴ A ∩ (B ∩ C) = {5, 7, 9} ∩ {9, 11} = {9}

A ∩ B = {5, 7, 9} ∩ {7, 9, 11} = {7, 9}

A ∩ C = {5, 7, 9} ∩ {9, 11} = {9}

∴ (A ∩ B) ∩ (A ∩ C) = {7, 9} ∩ {9} = {9}

Hence, A ∩ (B ∩ C) = (A ∩ B) ∩ (A ∩ C)

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